There has been great interest in two-dimensional representations of Galois groups, from conjectures of Artin concerning complex projective representations of the symmetries of the Platonic solids, to conjectures of Shimura and Taniyama concerning p-adic representations associated to elliptic curves. Many of these conjectures were recently answered in the affirmative by Wiles and Taylor using techniques from arithmetic algebraic geometry. In this talk, we explain how these results can be extended even further, and give some applications.